Find the triangles with angles A, B, CA,B,C and correspondingly opposite sides a,b.ca,b.c such that (aA+bB+cC)/(a+b+c)aA+bB+cCa+b+c has a minimum. Does this expression have a maximum?
2 Answers
The minimum would be zero. The maximum is infinity.
Explanation:
As given in the problem, the sides could shrink to a point, regardless of the angles. Similarly, there is no bound on the length of any of the sides.
For example, an equilateral triangle could be expanded infinitely. The ratio would continue to increase.
See below.
Explanation:
Considering the Chevishef's inequality
for
we can arrange
so we have
and then
The maximum determination (not attainable) is left as an exercise.